Matrix algebra and its applications to statistics and econometrics.

*(English)*Zbl 0915.15001
Singapore: World Scientific Publishing. xx, 535 p. (1998).

This book is designed to be a self-contained guide to matrix theory and linear algebra as used in statistics and econometrics, directed at graduate students and research workers both in mathematics and in the fields of applications of it. Mathematicians may also want to acquire it as a reference. It covers a great deal of interesting material rather quickly, far more than most books in linear algebra.

Chapter 1 deals with topics like linear independence, basis, dual space. Chapter 2 deals with inner products leading up to spectral theory for Hermitian conjugate bilinear functionals. Chapter 3 passes from linear transformations to their matrices. Chapter 4 treats rank and determinants.

Chapter 5 gives factorizations such as upper-lower triangular, QR factorization, Gram-Schmidt orthogonalization, Cholesky decomposition. In all 23 results on different kinds of factorizations are given. Chapter 6 deals with Kronecker products, Hadamard-Schur products, Khatri-Rao products, and calculus with matrices. Chapter 7 is on idempotents, Chapter 8 on generalized inverses and their relationship to regression.

Chapter 9 treats majorization of vectors from the viewpoint of doubly stochastic matrices. Chapter 10 gives eigenvalue inequalities such as Monotonicity Theorem, Interlace Theorem, PoincarĂ© Separation Theorem, Courant-Fischer Theorem, Horn’s Theorem, Von Neumann Theorem. Chapter 11 studies matrix norms, additional inequalities, and best approximations of matrices by other matrices of special types. Chapter 12 studies best estimation of linear statistical models such as are used in econometrics.

Chapter 13 outlines the theory of quadratic subspaces with applications to block designs. Chapter 14 is on inequalities of Cauchy-Schwarz, Hadamard, information theory, Jensen, Kantorovich, and related inequalities. Chapter 15 gives the Perron-Frobenius theory of nonnegative matrices and applications to economics and genetics. Chapter 16 gives some additional topics such as Toeplitz matrices and circulants.

I recommend this book for its extensive coverage of topics not easily found elsewhere and for its focus on applications.

Chapter 1 deals with topics like linear independence, basis, dual space. Chapter 2 deals with inner products leading up to spectral theory for Hermitian conjugate bilinear functionals. Chapter 3 passes from linear transformations to their matrices. Chapter 4 treats rank and determinants.

Chapter 5 gives factorizations such as upper-lower triangular, QR factorization, Gram-Schmidt orthogonalization, Cholesky decomposition. In all 23 results on different kinds of factorizations are given. Chapter 6 deals with Kronecker products, Hadamard-Schur products, Khatri-Rao products, and calculus with matrices. Chapter 7 is on idempotents, Chapter 8 on generalized inverses and their relationship to regression.

Chapter 9 treats majorization of vectors from the viewpoint of doubly stochastic matrices. Chapter 10 gives eigenvalue inequalities such as Monotonicity Theorem, Interlace Theorem, PoincarĂ© Separation Theorem, Courant-Fischer Theorem, Horn’s Theorem, Von Neumann Theorem. Chapter 11 studies matrix norms, additional inequalities, and best approximations of matrices by other matrices of special types. Chapter 12 studies best estimation of linear statistical models such as are used in econometrics.

Chapter 13 outlines the theory of quadratic subspaces with applications to block designs. Chapter 14 is on inequalities of Cauchy-Schwarz, Hadamard, information theory, Jensen, Kantorovich, and related inequalities. Chapter 15 gives the Perron-Frobenius theory of nonnegative matrices and applications to economics and genetics. Chapter 16 gives some additional topics such as Toeplitz matrices and circulants.

I recommend this book for its extensive coverage of topics not easily found elsewhere and for its focus on applications.

Reviewer: Ki Hang Kim (Montgomery)

##### MSC:

15-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to linear algebra |

62Jxx | Linear inference, regression |

62P20 | Applications of statistics to economics |